Conventionally, hot-press forming is widely known in which a pressing machine to which a die consisting of an upper die and a lower die is fixed presses a workpiece, such as a steel plate, heated to above a temperature at which an austenite structure appears, and at the same time, quenches the workpiece utilizing cooling by contact between the die and the workpiece.
A technique for the hot-press forming is publicly known which enables the die to suitably cool the workpiece during the quenching by providing water channels through which cooling water to cool the die flows in the die (for example, see Patent Literature 1).
However, since the pressed workpiece slightly deforms because of wrinkles thereof, spring back and the like, a gap is formed between the workpiece and the die when the die is closed. Consequently, a contact area between the surface of the workpiece and the forming surface of the die decreases during the quenching, which causes a problem that hardness of some parts in the workpiece is smaller than a predetermined value because of an insufficient cooling rate.
With reference to FIG. 7, described below is a problem occurring when a pressing machine (not shown) to which as die 100 as a conventional hot-pressing die is fixed performs the hot-press forming of a workpiece W formed in a flat plate to produce a product P.
For convenience, a top-bottom direction in FIG. 7 is defined as a top-bottom direction of the die 100, and a right-left direction in FIG. 7 is defined as a right-left direction of the die 100. In addition, this side in FIG. 7 is defined as a front side of the die 100, and the far side in FIG. 7 is defined as a rear side of the die 100, thereby a front-rear direction of the die 100 being defined.
As shown in FIG. 7, the die 100 includes a lower die 101 and an upper die 102 which are arranged so that the forming surfaces thereof are opposed to each other.
The lower die 101 has a protrusion 101a which protrudes upward from the intermediate part (the substantially middle part) of the forming surface thereof in the right-left direction.
The upper die 102 has a recess 102a in which the intermediate part (the substantially middle part), in the right-left direction, of the forming surface thereof dents upward along the shape of the protrusion 101a. 
The lower die 101 and the upper die 102 are arranged so that the forming surfaces thereof are opposed to each other. The upper die 102 approaches the lower die 101 to move to the bottom dead center, and thereby the lower die 101 and the upper die 102 perform the hot-press forming of the workpiece W arranged therebetween to produce the product P.
The product P has a shape in which what is called a hat-shaped section taken along the forming surfaces of the lower die 101 and the upper the 102 continues in the front-rear direction. The longitudinal direction of the product P corresponds to the front-rear direction.
Specifically, the product P has a top part P1 extending in the right-left direction in the uppermost part of the product P, two lateral parts P2 extending downward from both the ends of the top part P1 in the right-left direction, and two flanges P3 extending outward in the right-left direction from the bottom ends of the lateral parts P2.
In the product P with the above-mentioned shape, the top part P1 comes out of proximity with the protrusion 101a of the lower die 101 after the press working because of spring back and the like, and thereby a gap is formed between the product P and the die 100. Consequently, some parts in the product P are cooled at an insufficient cooling rate during the quenching. In particular, the substantially middle part of the top part P1 in the right-left direction has such a tendency, and the hardness thereof may be insufficient.
Pressing the workpiece at a relatively high pressure can solve the above-mentioned problem. However, it is disadvantageous in that an increase in size of the press machine for performing the hot-press forming leads to an increase in cost and the like.